There are four wheels that will give you a 7-pointed figure when used by the 150/105 ring: the 30, 45, 60 and 75.

Here they are combined in one design, using the new Spirograph Deluxe Set and the red, green and blue pens that came with it.

How to make it

I started with the smallest 30-tooth wheel, using holes 1, 2 and 3 with the green pen, to make the outermost set of patterns. Remember that you have to line the holes up carefully with a mark on the wheel before beginning to draw each pattern.

Then I took the 45 wheel, skipped the #4 hole and drew with holes 5, 6 and 7 with blue.

With the 60 wheel, I jumped ahead to 9, 10 and 11 using the red pen.

Finally I used wheel 75 to draw the final set using holes 13, 14 and 15 in green again.

Try it!

The math part

Notice how these four wheels, 30, 45, 60 and 75, are all multiples of 15!

If there were a tiny 15-tooth wheel, it would form a heptagon, without any crossing lines, because the inside of the 150/105 ring has 105 teeth, and 105 divided by 15 is 7. So the 15-toothed wheel would go around 7 times inside the 105-toothed ring before it arrived at the starting point.

Wheel 30 crosses one point place on its way to the next point, and goes around twice to meet the starting point.

Wheel 45 crosses two points and goes around 3 times to get back to the starting point.

Wheel 60 crosses 3 points, and wheel 75 crosses 4 points.

Trace these patterns with your finger or mouse in the design above and you’ll see it.